Desuspension Explained

In topology, a field within mathematics, desuspension is an operation inverse to suspension.^[1]

Definition

In general, given an n-dimensional space

X , the suspension \Sigma{X} has dimension n + 1. Thus, the operation of suspension creates a way of moving up in dimension. In the 1950s, to define a way of moving down, mathematicians introduced an inverse operation \Sigma ^{-1} , called desuspension.^[2] Therefore, given an n-dimensional space X , the desuspension \Sigma ^{-1} {X} has dimension n – 1.

In general,

\Sigma ^{-1} \Sigma{X}\ne X .

Reasons

The reasons to introduce desuspension:

1. Desuspension makes the category of spaces a triangulated category.
2. If arbitrary coproducts were allowed, desuspension would result in all cohomology functors being representable.

See also

• Cone (topology)
• Equidimensionality
• Join (topology)

External links

• Desuspension at an Odd Prime
• When can you desuspend a homotopy cogroup?

Notes and References

1. Luke. Wolcott. Elizabeth. McTernan. Imagining Negative-Dimensional Space. 637–642. Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture. 2012. Robert. Bosch. Douglas. McKenna. Reza. Sarhangi. 978-1-938664-00-7. 1099-6702. Tessellations Publishing. Phoenix, Arizona, USA. 25 June 2015. 26 June 2015. https://web.archive.org/web/20150626111631/http://bridgesmathart.org/2012/cdrom/proceedings/65/paper_65.pdf. dead.
2. Book: Margolis. Spectra and the Steenrod Algebra. North-Holland. 1983. 978-0-444-86516-8. North-Holland Mathematical Library. 454. 83002283.